On the linear-type property of the Jacobian ideal of affine plane curves (Q6595544)

Let \(I\) be an ideal of a commutative Noetherian ring \(R\). If the natural surjection from the Rees algebra of \(I\) onto the symmetric algebra of \(I\) is also injective, it is said that \(I\) is of linear type. Now assume that \(k\) is an algebraically closed field of characteristic zero and let \(f\) be a reduced polynomial (that is, in the factorization of \(f\) all factors have multiplicity 1) in \(S=k[x,y]\). Then \(X=V(f)\) is a reduced plane curve and the ideals \(J(f)=\langle \partial f/\partial x, \partial f/\partial y \rangle\) and \(I(f)=\langle f, J(f) \rangle\) are called the gradient and the Jacobian ideal of \(X\), respectively. It is said that \(X\) is of Jacobian linear type, when the Jacobian ideal of \(X\) is of linear type. In this paper, an equivalent condition for being of Jacobian linear type is presented for a reduced plane curve and then it is used to\N\begin{itemize}\N\item[1.] prove that every reduced plane curve with singular points of multiplicity 2 is of Jacobian linear type;\N\item[2.] characterize reduced plane curves with a singular point of multiplicity 3.\N\end{itemize}\NThe condition on \(X\) that the authors prove to be equivalent with being of Jacobian linear type, is being locally Eulerian, that is, for every singular point \(p\) of \(X\), it holds that \(f\in J(f)_{\mathfrak{m}_p}\), where \(\mathfrak{m}_p\) is the maximal ideal of \(S\) corresponding to \(p\). At the end of the paper, the authors state as a conjecture a characterization of reduced plane curves with singular points of multiplicity 4.